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Uniform convergence of double trigonometric series. (English) Zbl 0849.42008

The authors consider double trigonometric series \[ \sum^\infty_{j= -\infty} \sum^\infty_{k= -\infty} c_{jk} e^{i(jx+ ky)},\tag{\(*\)} \] where the \((c_{jk})\) are complex numbers. They give sufficient conditions under which the rectangular partial sums \(s_{mn}(x, y)\) of \((*)\) converge uniformly on the two-dimensional torus. These conditions include that \((c_{jk})\) is a double null sequence and of bounded variation of order \((1, 0)\), \((0, 1)\), and \((1, 1)\) with the weights \(|j|\), \(|k|\), and \(|jk|\), respectively.
The results of the present paper generalize those by J. R. Nurcombe [J. Math. Anal. Appl. 166, No. 2, 577-581 (1992; Zbl 0756.42006)] and T. F. Xie and S. P. Zhou [J. Math. Anal. Appl. 181, No. 1, 171-180 (1994; Zbl 0791.42004)].
Reviewer: F.Móricz (Szeged)

MSC:

42B08 Summability in several variables
42B05 Fourier series and coefficients in several variables
42A20 Convergence and absolute convergence of Fourier and trigonometric series
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